The Generating Idempotent Is a Minimum-Weight Codeword for Some Binary BCH Codes

TL;DR

This paper proves a conjecture from 2015 that the generating idempotent of certain binary BCH codes has weight equal to the Bose distance, using properties of fibbinary integers and related polynomials.

Contribution

It establishes that the generating idempotent's weight equals the Bose distance for both odd and even m in binary BCH codes, confirming a longstanding conjecture.

Findings

Proves the conjecture for odd m

Shows the weight of the generating idempotent equals the Bose distance for all m

Uses properties of fibbinary integers and polynomial relations

Abstract

In a paper from 2015, Ding et al. (IEEE Trans. IT, May 2015) conjectured that for odd $m$, the minimum distance of the binary BCH code of length $2^m-1$ and designed distance $2^{m-2}+1$ is equal to the Bose distance calculated in the same paper. In this paper, we prove the conjecture. In fact, we prove a stronger result suggested by Ding et al.: the weight of the generating idempotent is equal to the Bose distance for both odd and even $m$. Our main tools are some new properties of the so-called fibbinary integers, in particular, the splitting field of related polynomials, and the relation of these polynomials to the idempotent of the BCH code.